Abstract
We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in \({\mathbb{R}^{n}}\) and then for “abstract” cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in \({\mathbb{R}^{n}}\) .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fulton W., Sturmfels B.: Intersection theory on toric varieties. Topology 36(2), 335–353 (1997)
Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compositio Mathematica (to appear). preprint math.AG/0708.2268
Gathmann A., Markwig H.: Kontsevich’s formula and the WDVV equations in tropical geometry. Adv. Math. 217(2), 537–560 (2008)
Katz, E.: A tropical toolkit. preprint math.AG/0610878
Kerber, M., Markwig, H.: Counting tropical elliptic plane curves with fixed j-invariant. Commentarii Mathematici Helvetici (to appear). preprint math.AG/0608472
Mikhalkin, G.: Tropical Geometry and its applications. In: Proceedings of the ICM, Madrid, Spain, pp. 827–852 (2006)
Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Litvinov, G.L., Maslov, V.P. (eds.) Idempotent Mathematics and Mathematical Physics. Proceedings Vienna 2003, American Mathematical Society, Contemp. Math., pp. 377 (2005)
Sturmfels B., Tevelev J.: Elimination theory for tropical varieties. Math. Res. Lett. 15(3), 543–562 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Allermann, L., Rau, J. First steps in tropical intersection theory. Math. Z. 264, 633–670 (2010). https://doi.org/10.1007/s00209-009-0483-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-009-0483-1